Mathematics of the Wankel Engine shapes

Regarding the mathematics of the unusual shape and profile of the Wankel engine triangular rotor and combustion chamber housing, I'm reviewing a mathematics demonstration I have just come across but which has been on the internet for a few years but, like me, you may not have come across it before now.

This Demonstration gives an animation of an epitrochoid and associated planetary-motion envelope curve. The configuration shown has applications in the internal combustion engines invented by Felix Wankel and popularized by Mazda in RX-7 and RX-8 cars

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Review by Peter Dow

If you think this video looks interesting, I highly recommend that you download the Wolfram CDF player software so that you can experiment with the features of Tony Kelman's demonstration. To quote Tony

"The "eccentricity ratio" changes the shapes of the curves. The "reference frame" determines what is held stationary in the animation: either the epitrochoid (blue), the envelope curve (purple), or the centers of rotation of both curves. The "inner" envelope is the triangular rotor shape used in place of a piston in a Wankel rotary engine, whereas the "outer" envelope is the continuation of the envelope curve along the opposite extreme of motion."

Looking at eccentricity ratios widely different from what we see in real Wankel engines is quite a revelation too.

As if all that wasn't enough, you also get to download and look at Tony's open source code and in particular the maths equations he uses to generate the curves.

Tony suggests some extensions to his demonstration. Well I have ideas of my own - I'd like to see computations of the areas between the curves representing the combustion chambers and a calculation of compression ratios for example.

Unfortunately, I don't have the Mathematica developers software package which, unlike the free player I got to view the demo, you have to pay - A LOT - for.

As part of my mechanical engineering degree I have just completed my final project on a computational fluid dynamics model of the Wankel Rotary Engine. This has involved a lot of geometric work on trochoids and their different variations. I have used the Wolfram tool you have described above.

In regards to the areas between the curves there is some literature deriving equations to calculate volumes / areas at particular rotor positions, however I found these differed slightly from volumes taken directly from CAD data.

Also, an important factor is the clearance height between rotor and housing. In reality the housing curve is offset by an amount to give a clearance between rotor and housing. This distance varies depending on the rotor position and is taken up by the spring loaded apex seals. This offset distance has a significant effect on the volume at minimum and maximum positions and therefore compression ratio.
For example my engine had an eccentricity of 6.6mm and a rotor radius of 48mm. Using theoretical calculations in the literature this gave a compression ratio of 18.5. However a clearance height of only 1mm reduced the CR to 10.

Another thing to note is the effect of the ports. For a side ported design the ports open progressively so calculating compression ratio is not as simple as taking the ratio of maximum and minimum volumes.

"Rotary Engine" by Kenichi Yamamoto evaluates the mathematical principles of the Wankel engine quite well.
Also "Wankel RC Engine" by R.F. Ansdale has a quite in-depth review of this.

More recently Warren and Yang published a paper "Design of rotary engines from the apex seal profile" which used a modified housing profile which is always perpendicular to the apex seal to reduce leakage.

Thank you for your reply Kieran. Yes that did help and I may wish to get back to you later with specific questions if you would like to help further?

After I started this topic, I have since found another trochoids interactive demonstration webpage, this time by Christopher J. Henrich. His code is in Javascript which means it is pretty much open source, can run on most modern web browsers and therefore is ideal for me to modify.

So I've made a start and I'm publishing a webpage today which partially performs some of what Tony Kelman's demonstration does. I've a lot more to do yet but if you want to see how far I've got and monitor my progress, then click to my webpage using the following link.

My page includes links back to Christopher J. Henrich's original webpage and he is OK with me publishing this link. Anyway see for yourself.

"Rotary Engine" by Kenichi Yamamoto evaluates the mathematical principles of the Wankel engine quite well.
Also "Wankel RC Engine" by R.F. Ansdale has a quite in-depth review of this.

Click to expand...

Do you have a copy of Ansdale to hand Kieran? If so I wonder if you'd consider scanning some pages of it in and sending them to me as I suggest below?

Rotary engine rotor-contour equations

Tony Kelman in his "Wankel Rotary Engine: Epitrochoidal Envelopes" demonstration describes the curve representing the contour of the rotor as an "envelope" and that's likely to be because he based his Mathematica program code which draws that rotor "envelope" curve on an "inner envelope" equation which he has referenced as derived in a book by R. F. Ansdale, "The Wankel RC Engine: Design and Performance", published in New York: by A. S. Barnes in 1969.

According to several reviews I've read, Ansdale's book is a valuable reference for the mathematics of the Wankel rotary engine and other rotary machines but, somewhat frustratingly, unlike Yamamoto's "Rotary Engine" book which one can download as a pdf file for free, Ansdale's book is not yet freely available for download on the internet, so most of what Ansdale writes there remains out of reach to me to me at this time. If anyone out there has a copy of Ansdale within reach and is willing to scan in certain pages of that book which I am keen to read and then email the images to me, I'd be grateful for the help.

Wanted: Image scans of Ansdale, pages 136 to 139 plus the part of the appendix which Tony Kelman writes about in this quote "The appendix of the Ansdale book has the equations derived for arbitrary numbers of lobes, if I remember correctly."

So that would be most relevant information to help me develop my webpage and so if anyone has hands on access to a copy of Ansdale then please scan and email those pages to me at peterdow@talk21.com. Thanks!

Inner envelope equation for the 2-lobe peritrochoid

Tony Kelman was kind enough to take the time to help me to compare the rotor envelope equation he had used with the equations which Yamamoto had published for the same thing.

I say "equations", plural, because we noticed a critical typographical mistake in Yamamoto's 1971 edition, misstating an inaccurate envelope equation - which typo Yamamoto had corrected in his 1981 edition of "Rotary Engine", as described in this "Compare the TYPO!" image.

Equivalence of Ansdale-1969 & Yamamoto-1981 inner envelope equations

At first glance it may not look like it but mathematically the following two equations - Ansdale's and Yamamoto's - are indeed equivalent - they both draw exactly the same curve when plotted and give numerically identical values to the available precision.

The equivalence of these equations is analytically demonstrated below by replacing the coloured terms of say Ansdale's equation with their similarly coloured equivalent terms in Yamamoto's equation, by rearranging the terms or by using standard trigonometric product-to-sum identities, listed below.

Here are the those equations again but posted as message text. All that seems to be missing is the overline bar for the square root symbol because I don't yet know how to format an overline on vBulletin. The way an overline is formatted in HTML is as follows

HTML:

<span style="text-decoration: overline">text to have an overline over it</span>

Ansdale - 1969
Adjust some of the terms in Ansdale's equation by multiplying by 1 expressed as "Â½ Ã— 2" and arrange the factors to allow the identification of equivalent factors in both equations.

Yamamoto - 1981
Adjust some of the terms in Yamamoto's equation by multiplying some terms by 1 expressed as "R Ã— (1/R)" and then express the equation as terms which are factors each of which is multiplied by the common factor of R. Also, rearrange the 2nd X term which Yamamoto has as an addition term with a "(cos 8v - cos 4v)" trig factor by multiplying the term by 1 performed as multiplication by "-1 Ã— -1", changing the addition term to a minus term and swapping the trig factor subtraction minuend and subtrahend noting that "-1 Ã— (cos 8v - cos 4v) = (cos 4v - cos 8v)".